Putnam 1948 A6

Answer either (i) or (ii):

(i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.

(ii) Show that $$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$